Finite automata became over the years the standard representation of finitely generated subgroups $H$ of a free group $F_A$. The Stallings construction constitutes a simple and efficient algorithm for building an automaton $S(H)$ which can be used for solving the membership problem of $H$ in $F_A$ and many other applications. This automaton $S(H)$ is nothing more than the core automaton of the Schreier graph (automaton) of $H$ in $G$, whose structure can be described as $S(H)$ with finitely many infinite trees adjoined.

Such an approach invites naturally generalizations for further classes of groups. For instance, an elegant geometric construction of Stallings type automata was achieved for amalgams of finite groups by Markus-Epstein. On the other hand, the most general results were obtained by Kapovich, Weidmann and Miasnikov for finite graphs of groups where each vertex group is either polycyclic-by-finite or word-hyperbolic and locally quasiconvex, and where all edge groups are virtually polycyclic. However, the complex algorithms were designed essentially to solve the generalized word problem, and it seems very hard to extend other features of the free group case, either geometric or algorithmic.

In joint work with Xaro Soler-Escriv\'a and Enric Ventura, we present a Stallings type approach with some generality which is robust enough to exhibit several prized algorithmic and geometric features, namely in connection with Schreier graphs. Moreover, using Bass-Serre theory, we succeed on identifying those groups $G$ for which it can be carried on: (finitely generated) virtually free groups.